As shown in FIG. 1, a prior art bicycle handlebar upright tube 93 is provided at one end thereof with slot 94 having in the midsegment thereof a slot hole 95 corresponding in location to and engageable with two parallel planar surfaces 92 located at the top of a bicycle head tube 91. The handlebar upright tube 93 and the bicycle head tube 91 can be fastened or unfastened by means of a fastening member 98 so as to allow a bicyclist to adjust the angular position of the handlebar upright tube 93 without the help of a hand tool. However, such a prior art means of fastening the handlebar upright tube 93 with the head tube 91 as described is defective in design in that the construction of the two parallel planar surfaces 92 of the head tube 91 and the construction of the slot hole 95 of the handlebar upright tube 93 are rather complicated and are therefore not cost-effective. In addition, the construction of the fastening member 98 is such that it can be interfered with by a foreign object.